Journal of the
Korean Mathematical Society JKMS

Weyl structures can be viewed as generalizations of Riemannian metrics. We study Weyl structures which are critical points of the squared $L^2$ norm functional of the full curvature tensor, defined on the space of Weyl structures on a compact 4-manifold. We find some relationship between these critical Weyl structures and the critical Riemannian metrics. Then in a search for homogeneous critical structures we study left-invariant metrics on some solv-manifolds and prove that they are not critical.

Keywords: critical Weyl structure, Einstein-Weyl structure, quadratic geometric functional, solv-manifold

MSC numbers: Primary 58E11; Secondary 53C25